minimal polynomial
Let be a field extension and be algebraic over . The minimal polynomial for over is a monic polynomial such that and, for any other polynomial with , divides . Note that, for any element that is algebraic over , a minimal polynomial exists (http://planetmath.org/ExistenceOfTheMinimalPolynomial); moreover, because of the monic condition, it exists uniquely.
Given , a polynomial is the minimal polynomial of if and only if and is both monic and irreducible (http://planetmath.org/IrreduciblePolynomial).